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Chapter 5: Problem 75

Coin If you flip a fair coin repeatedly and the first four results are tails, are you more likely to get heads on the next flip, more likely to get tails again, or equally likely to get heads or tails?

Short Answer

Expert verified
The likelihood to get heads or tails on the next flip after four consecutive tails is equal.

Step by step solution

01

Understand Independence of Events

The most important fundamental concept in this problem is that coin flips are independent events. That is, the result of one coin flip does not influence or change the result of any other coin flip. The coin does not have a memory and does not keep track of the previous results.
02

Calculate the Probability

A fair coin has two outcomes: heads and tails. Each of these outcomes has an equal chance of occurring on each flip. This is represented mathematically as \(P(\text{Heads}) = \frac{1}{2}\) and \(P(\text{Tails}) = \frac{1}{2}\).
03

Apply These Concepts to the Exercise

Since every coin flip is independent, the result of the fifth coin flip is not influenced by the results of the first four. This means the likelihood of getting heads or tails on the next flip remains equal.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Independent Events
In probability, independent events are those where the outcome of one event does not affect the outcome of another. Coin tosses are a classic example of independent events because each flip is entirely unaffected by previous results.
Imagine you flip a coin. Whether it lands heads or tails does not impact the next toss. The coin does not "remember" past results, allowing each flip to have the same probability of heads or tails.
It's crucial to understand that independence means every trial is isolated - no cumulative memory or connection to past outcomes.
Fair Coin
A fair coin is one that has no inclination to land more on one side than the other. This means that the chances of obtaining heads or tails on a flip are equal, each being 50%.
In practice, a fair coin would show heads and tails each time it is tossed, evenly distributed over many tosses.
The fairness ensures an unbiased random event, which is important in probability experiments to maintain accuracy and reliability.
Outcomes of a Coin Flip
The outcomes of a coin flip refer to the possible results: either heads or tails. Each flip of a coin results in one of these two outcomes.
If the coin is fair, as discussed, the probability of each outcome is equal. Mathematically, this can be written as \(P(\text{Heads}) = \frac{1}{2}\) and \(P(\text{Tails}) = \frac{1}{2}\).
Understanding the possible outcomes and their probabilities is vital when calculating and predicting results in probability-based exercises.
Randomness
Randomness is the lack of pattern or predictability in events. In the context of a coin flip, randomness implies that each flip is independent and unpredictable.
This is central to probability theory, as randomness ensures that no certain patterns dictate the outcomes over time.
  • It allows us to predict outcomes based on probabilities rather than past sequences.
  • While patterns might seem to emerge over many flips, each new toss does not "depend" on previous ones.
Understanding randomness helps to grasp the unbiased nature of probability events.

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Most popular questions from this chapter

Use your general knowledge to label the following pairs of variables as independent or associated. Fxplain. a. The outcome on flips of two separate, fair coins b. Breed of dog and weight of dog for dogs at a dog show

Let \(H\) stand for heads and let \(T\) stand for tails in an experiment where a fair coin is flipped twice. Assume that the four outcomes listed are equally likely outeomes: $$ \mathrm{HH}, \mathrm{HT}, \mathrm{TH}, \mathrm{TT} $$ What are the probabilities of getting the following: a. 0 heads b. Exactly 1 head c. Lixactly 2 heads d. At least one 1 head e. Not more than 2 heads

In 2017 the Pew Research Center asked young adults aged 18 to 29 about their media habits. When asked, "What is the primary way you watch television?" \(61 \%\) said online streaming service, \(31 \%\) said cable/satellite subscription, and 5\% said digital antenna. Suppose the Pew Research Center polled another sample of 2500 young adults from this age group and the percentages were the same as those in 2017 . a. How many would say online streaming services? b. How many would say cable/satellite subscription? c. How many would say cable/satellite subseription or digital antenna? d. Are the responses "online streaming service," "cable/satellite subscription," and "digital antenna" mutually exclusive? Why or why not?

In order to practice law, lawyers must pass the bar exam. In California, the passing rate for first-time bar exam test takers who attended an accredited California law school was \(70 \%\). Suppose two test-takers from this group are selected at random. a. What is the probability that they both pass the bar exam? b. What is the probability that only one passes the bar exam? c. What is the probability that neither passes the bar exam?

a. Explain how you could use a random number table (or the random numbers generated by software or a calculator) to simulate rolling a fair four-sided die 20 times, Assume you are interested in the probability of rolling a \(1 .\) 'Then report a line or two of the random number table (or numbers generated by a computer or calculator) and the values that were obtained from it. b. Report the empirieal probability of rolling a 1 on the four-sided die from part (a), and compare it with the theoretical probability of rolling a 1 .
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